timsac (version 1.2.7)

mulbar: Multivariate Bayesian Method of AR Model Fitting

Description

Determine multivariate autoregressive models by a Bayesian procedure. The basic least squares estimates of the parameters are obtained by the householder transformation.

Usage

mulbar(y, max.order=NULL, plot=FALSE)

Arguments

y

a multivariate time series.

max.order

upper limit of the order of AR model. Default is $2 \sqrt{n}$, where $n$ is the length of the time series y.

plot

logical. If TRUE daic is plotted.

Value

• mean
  mean.
• var
  variance.
• v
  innovation variance.
• aic
  AIC.
• aicmin
  minimum AIC.
• daic
  AIC-aicmin.
• order.maice
  order of minimum AIC.
• v.maice
  MAICE innovation variance.
• bweight
  Bayesian weights.
• integra.bweight
  integrated Bayesian Weights.
• arcoef.for
  AR coefficients (forward model). arcoef.for[i,j,k] shows the value of $i$-th row, $j$-th column, $k$-th order.
• arcoef.back
  AR coefficients (backward model). arcoef.back[i,j,k] shows the value of $i$-th row, $j$-th column, $k$-th order.
• pacoef.for
  partial autoregression coefficients (forward model).
• pacoef.back
  partial autoregression coefficients (backward model).
• v.bay
  innovation variance of the Bayesian model.
• aic.bay
  equivalent AIC of the Bayesian (forward) model.

Details

The statistic AIC is defined by $$AIC = n\ log(det(v)) + 2k,$$ where $n$ is the number of data, $v$ is the estimate of innovation variance matrix, $det$ is the determinant and $k$ is the number of free parameters. Bayesian weight of the $m$-th order model is defined by $$W(n) = const \times C(m) / (m+1),$$ where $const$ is the normalizing constant and $C(m)=exp(-0.5 AIC(m))$. The Bayesian estimates of partial autoregression coefficient matrices of forward and backward models are obtained by $(m = 1,\ldots,lag)$ $$G(m) = G(m) D(m),$$ $$H(m) = H(m) D(m),$$ where the original $G(m)$ and $H(m)$ are the (conditional) maximum likelihood estimates of the highest order coefficient matrices of forward and backward AR models of order $m$ and $D(m)$ is defined by $$D(m) = W(m) + \ldots + W(lag).$$ The equivalent number of parameters for the Bayesian model is defined by $$ek = (D(1)^2 + \ldots + D(lag)^2) id + id (id+1)/2$$ where $id$ denotes dimension of the process.

References

H.Akaike (1978) A Bayesian Extension of The Minimum AIC Procedure of Autoregressive Model Fitting. Research Memo. NO.126, The Institute of Statistical Mathematics. G.Kitagawa and H.Akaike (1978) A Procedure for The Modeling of Non-stationary Time Series. Ann. Inst. Statist. Math., 30, B, 351--363. H.Akaike, G.Kitagawa, E.Arahata and F.Tada (1979) Computer Science Monograph, No.11, Timsac78. The Institute of Statistical Mathematics.

Examples

data(Powerplant)
  z <- mulbar(Powerplant, max.order=10)
  z$pacoef.for
  z$pacoef.back